Maximal branching processes in random environment

The work continues the author's long research in the theory of maximal branching processes that are obtained from classical branching processes by replacing the sum of offsping numbers by the maximum. One can say that the next generation is formed by the offspring of the most productive particle. Earlier, the author generalized processes with integer values up to processes with arbitrary nonnegative values, investigated their properties, and proved the limit theorems. Further, maximal branching processes with several types of particles were introduced and studied. In this paper, the author introduces the concept of maximal branching processes in random environment (with one type of particles) and an important case of the “power” random environment. In the latter case, the basic properties of maximal branching processes are studied and the ergodic theorem is proved. As an application, the author considers gated infinite-server queues.